Theorem conncompss 23457

Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypothesis

Ref	Expression
conncomp.2	⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}

Assertion

Ref	Expression
conncompss	⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑆)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋

Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem conncompss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1135	. . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑋)
2		conntop 23441	. . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Conn → (𝐽 ↾t 𝑇) ∈ Top)
3	2	3ad2ant3 1134	. . . . . 6 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝐽 ↾t 𝑇) ∈ Top)
4		restrcl 23181	. . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V))
5	4	simprd 495	. . . . . 6 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → 𝑇 ∈ V)
6		elpwg 4608	. . . . . 6 ⊢ (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
7	3, 5, 6	3syl 18	. . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
8	1, 7	mpbird 257	. . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ∈ 𝒫 𝑋)
9		3simpc 1149	. . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn))
10		eleq2 2828	. . . . . 6 ⊢ (𝑦 = 𝑇 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑇))
11		oveq2 7439	. . . . . . 7 ⊢ (𝑦 = 𝑇 → (𝐽 ↾t 𝑦) = (𝐽 ↾t 𝑇))
12	11	eleq1d 2824	. . . . . 6 ⊢ (𝑦 = 𝑇 → ((𝐽 ↾t 𝑦) ∈ Conn ↔ (𝐽 ↾t 𝑇) ∈ Conn))
13	10, 12	anbi12d 632	. . . . 5 ⊢ (𝑦 = 𝑇 → ((𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn) ↔ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn)))
14		eleq2 2828	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
15		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐽 ↾t 𝑥) = (𝐽 ↾t 𝑦))
16	15	eleq1d 2824	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐽 ↾t 𝑥) ∈ Conn ↔ (𝐽 ↾t 𝑦) ∈ Conn))
17	14, 16	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)))
18	17	cbvrabv 3444	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)}
19	13, 18	elrab2 3698	. . . 4 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn)))
20	8, 9, 19	sylanbrc 583	. . 3 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
21		elssuni 4942	. . 3 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
22	20, 21	syl 17	. 2 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
23		conncomp.2	. 2 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}
24	22, 23	sseqtrrdi 4047	1 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912  (class class class)co 7431   ↾_t crest 17467  Topctop 22915  Conncconn 23435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-rest 17469  df-top 22916  df-conn 23436

This theorem is referenced by:  conncompcld  23458  tgpconncompeqg  24136  tgpconncomp  24137